Modules over group rings of solvable groups with rank restrictions on subgroups

Let $A$ be an $\mathbf RG$-module over a commutative ring $\mathbf R$, where $G$ is a group of infinite section $p$-rank ($0$-rank), $C_G(A)=1$, $A$ is not a Noetherian $\mathbf R$-module, and the quotient $A/C_A(H)$ is a Noetherian $\mathbf R$-module for every proper subgroup $H$ of infinite section $p$-rank ($0$-rank). We describe the structure of solvable groups $G$ of this type.